Lagrange, along with Euler and the Bernoullis, developed the calculus of variations for dealing with mechanics. He was responsible for laying the groundwork for a different way of writing down Newton's Equations of Motion. This is what has been called Lagrangian Mechanics. It accomplishes the same thing that Newton's Equations provide - the path followed by all the bodies studied in a problem - but the form of the equations is actually better because the form does not change when the coordinate system used to describe a problem changes. Lagrangian Mechanics have been widely used to solve many mechanical problems.
He also introduced a new variable for studying conservative systems which is now called the Lagrangian. The integral of the Lagrangian of the system has been called the "action" of a system and is given the symbol S. This rewriting of Newton's equations opened minds to new forms of mechanics. However, though this rewrites Newton's Equations, there is nothing fundamentally different - it is simply a more convenient way to write it down and solve it.